A RAY THEORY FOR NONLINEAR SHIP WAVES 

 AND WAVE RESISTANCE 



Bohyun Yim 

 Taylor Naval Ship Research and Development Center 

 Bethesda, Maryland 20084, U.S.A. 



Abstract 



Analytical and numerical methods for 

 application of ray theory in computing ship 

 waves are investigated. The potentially 

 important role of ray theory in analyses of 

 nonlinear waves and wave resistance is demon- 

 strated. The reflection of ship waves from 

 the hull boundary is analyzed here for the 

 first time. 



When a wave crest touches the ship sur- 

 face, the ray exactly follows the ship surface. 

 When the wave crest is nearly perpendicular to 

 the ship surface the ray is reflected many 

 times as it propagates along the ship surface. 

 Many rays of reflected elementary waves 

 intersect each other. The envelope to the first 

 reflected rays forms a line like a shock 

 front which borders the area of large waves 

 or breaking waves near the ship. 



For the Wigley hull, ray paths, wave 

 phases, and directions of elementary waves 

 are computed by the ray theory and a method of 

 computing wave resistance is developed. The 

 wave phase is compared with that of linear 

 theory as a function of ship-beam length 

 ratio to Identify the advancement of the bow 

 wave phase which influences the design of bow 

 bulbs. The wave resistance of the Wigley hull 

 is computed using the amplitude function from 

 Michell's thin ship theory and compared with 

 values of Michell's wave resistance. The total 

 wave resistance has the phase of hump and hollow 

 shifted considerably. 



Introduction 



Significant developments in the ship wave 

 theory have been made in recent years. These 

 include the application of ray theoryl>2* an( j 

 the experimental discovery of a phenomenon 

 called a free surface shock wave. 3 



Because a ship is not thin enough to apply 

 both the thin ship theory and the complicated 

 free-surface effect, theoretical development 

 of an accurate ship wave theory has been slow. 

 The problem should be evaluated differently 

 from the conventional means. Ray theory has 

 been used in geometrical optics or for waves 

 having small wave lengths. Ursell used the 



*A complete listing of references is given 

 on page 1 5 . 



theory to consider wave propagation in non- 

 uniform flow, and Shen, Meyer, and Keller-* used 

 it to investigate water waves in channels and 

 around islands. Recently Keller developed a 

 ray theory for ship waves and pointed out that 

 the theory could supply useful information 

 about the waves of thick ships at relatively 

 slow speeds. However, he had difficulty ob- 

 taining the excitation function for wave ampli- 

 tude and solved only a thin-ship problem. Inui 

 and Kajitani^ applied the Ursell ray theory 4 to 

 ship waves, using the amplitude function from 

 linear theory. 



Because the ray is the path of wave energy, 

 it is not supposed to penetrate the ship surface; 

 and this was emphasized by Keller. However, 

 neither Keller nor Inui and Kajitani consider 

 nonpenetration of the ray seriously. Recently 

 Yim" found the existence of rays which emanated 

 from the ship bow and reflected from the ship 

 surface. 



In the present paper, further study has 

 revealed many bow rays reflecting from the 

 ship surface, creating an area in which these 

 rays intersect each other. The envelope to 

 the first reflected rays forms a line like a 

 shock front which borders the area of large 

 waves or breaking waves near the ship. This 

 will be referred to as the second caustic. This 

 phenomenon was observed by Inui et al.' in the 

 towing tank and called a free surface shock 

 wave (FSSW) . Much has been done concerning the 

 theoretical investigation of FSSW by Introduc- 

 ing a fictitious depth for a shallow water non- 

 dispersive wave. 



The ray equation and the ship boundary 

 condition are analyzed further to show the 

 existence of reflecting rays, except in the 

 case of a flat wedgelike ship surface. The 

 ray path is very sensitive to the initial bound- 

 ary condition near the bow or the stern, and 

 should be identified at downstream infinity 

 not by the initial condition. The conservation 

 law of wave energy in the nonuniform flow is 

 different from that of the uniform flow due to 

 the exchange of energy with the local flow. 8 

 Therefore, the linear wave amplitude as a 

 function of initial wave angle near the bow 

 or stern is meaningless and cannot be used in 

 the ray theory. It is shown that the amplitude 

 function for the ray theory matched with the 



